## Sloshing motions in excited tanks.(English)Zbl 1115.76369

Summary: A fully nonlinear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic base excitations are presented for small to steep non-breaking waves. The simulations are limited to a single water depth above the critical depth corresponding to a tank aspect ratio of $$hs/b=0.5$$. The numerical model is valid for any water depth except for small depth when viscous effects would become important. Solutions are limited to steep non-overturning waves. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and second order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the third-order single modal solution and the fully non-linear numerical model. The agreement is in general good, both amplitude and phase. As expected, the third-order compared to the second-order solution is more accurate. This is especially true for resonance, high forcing frequency and mode interaction cases. However, it was found that multimodal approximate forms should be used for the cases in which detuning effects occur due to mode interaction. We present some test cases where detuning effects are evident both for single dominant modes and mode interaction cases. Furthermore, for very steep waves, just before the waves overturn, and for large forcing frequency, a discrepancy in amplitude and phase occurs between the approximate forms and the numerical model. The effects of the simultaneous vertical and horizontal excitations in comparison with the pure horizontal motion and pure vertical motion is examined. It is shown that vertical excitation causes the instability associated with parametric resonance of the combined motion for a certain set of frequencies and amplitudes of the vertical motion while the horizontal motion is related to classical resonance. It is also found that, in addition to the resonant frequency of the pure horizontal excitation, an infinite number of additional resonance frequencies exist due to the combined motion of the tank. The dependence of the non-linear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem, non-linear effects become important when the steepness reaches about 0.1, in agreement with the physical experiments of Abramson [Rep. SP 106, NASA, 1966].

### MSC:

 76M20 Finite difference methods applied to problems in fluid mechanics

### Keywords:

Sloshing motion; Finite differences; Moving liquid tanks
Full Text:

### References:

 [1] Abramovitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover Publications New York [2] H.N. Abramson, The dynamics of liquids in moving containers, Rep. SP 106, NASA, 1966 [3] H.N. Abramson, R.L. Bass, O. Faltinsen, H.A. Olsen, Liquid slosh in lng carriers, in: 10th Symposium on Naval Hydrodynamics, Cambridge, Massachusetts, ACR-204, 1974, pp. 371-388 [4] Benjamin, T.B.; Ursell, F., The stability of the plane free surface of a liquid in a vertical periodic motion, Proc. R. soc. lond. ser. A, 225, 505-515, (1954) · Zbl 0057.18801 [5] Bredmose, H.; Brocchini, M.; Peregrine, D.H.; Thais, L., Experimental investigation and numerical modelling of steep forced water waves, J. fluid mech., 490, 217-249, (2003) · Zbl 1063.76501 [6] Cariou, A.; Casella, G., Liquid sloshing in ship tanks: a comparative study of numerical simulation, In marine struct., 12, 183-189, (1999) [7] Celebi, M.S.; Akyuldiz, H., Nonlinear modeling of liquid sloshing in moving rectangular tank, In Ocean eng., 29, 1527-1553, (2002) [8] Chen, W.; Haroun, M.A.; Liu, F., Large amplitude liquid sloshing in seismically excited tanks, Earthquake eng. struct. dyn., 25, 653-669, (1996) [9] Chern, M.J.; Borthwick, A.G.L.; Eatock Taylor, R., A pseudospectral σ-transformation model of 2-D nonlinear waves, J. fluids struct., 13, 607-630, (1999) [10] Faltinsen, O.M., A nonlinear theory of sloshing in rectangular tanks, J. ship res., 18, 4, 224-241, (1974) [11] Faltinsen, O.M.; Rognebakke, O.F.; Lukovsky, I.A.; Timokha, A.N., Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, J. fluid mech., 407, 201-234, (2000) · Zbl 0990.76006 [12] Faltinsen, O.M.; Rognebakke, O.F.; Timokha, A.N., Resonant three-dimensional nonlinear sloshing in a square-base basin, J. fluid mech., 487, 1-42, (2003) · Zbl 1053.76006 [13] Faltinsen, O.M.; Timokha, A.M., Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth, J. fluid mech., 470, 319-357, (2002) · Zbl 1163.76325 [14] Faraday, M., On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Phil. trans. R. soc. lond., 121, 299-340, (1831) [15] P. Ferrant, D. Le Touze, Simulation of sloshing waves in a 3D tank based on a pseudo-spectral method, in: Proc. 16th International Workshop on Water Waves and Floating Bodies, Hiroshima, Japan, 2001 [16] Frandsen, J.B.; Borthwick, A.G.L., Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver, J. fluids struct., 18, 2, 197-214, (2003) [17] J. Gerrits, Dynamics of liquid-filled spacecraft, PhD thesis, Rijks Universiteit Groningen, Holland, 2001 · Zbl 1112.76337 [18] Greaves, D.M.; Borthwick, A.G.L.; Wu, G.X.; Eatock Taylor, R., A moving boundary finite element method for fully non-linear wave simulations, J. ship res., 41, 3, 181-194, (1997) [19] Gu, X.M.; Sethna, P.R., Resonance surface waves and chaotic phenomena, J. fluid mech., 183, 543-565, (1987) · Zbl 0639.76015 [20] Gu, X.M.; Sethna, P.R.; Narain, A., On three-dimensional nonlinear subharmonic resonance surface waves in a fluid: part i – theory, J. appl. mech., 55, 213-219, (1988) [21] Hill, D.F., Transient and steady-state amplitudes of forced waves in rectangular basins, Phys. fluids, 15, 6, 1576-1587, (2003) · Zbl 1186.76228 [22] Hirt, C.W.; Nichols, B.D., Volume of fluid (vof) method for the dynamics of free boundaries, J. computat. phys., 39, 201-225, (1981) · Zbl 0462.76020 [23] Ibrahim, R.A.; Pilipchuk, V.N.; Ikeda, T., Recent advances in liquid sloshing dynamics, ASME appl. mech. rev., 54, 2, 133-199, (2001) [24] Jiang, L.; Perlin, M.; Schultz, W.W., Period tripling and energy dissipation of breaking standing waves, J. fluid mech., 369, 273-299, (1998) · Zbl 0955.76509 [25] Jiang, L.; Ting, C-L.; Perlin, M.; Schultz, W.W., Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries, J. fluid mech., 329, 275-307, (1996) · Zbl 0900.76014 [26] Kareem, A.; Kijewski, T.; Tamura, Y., Mitigation of motions of tall buildings with specific examples of recent applications, J. wind struct., 2, 3, 201-251, (1999) [27] Kocygit, M.B.; Falconer, R.A.; Lin, B., Three-dimensional numerical modelling of free surface flows with non-hydrostatic pressure, Int. J. numer. meth. fluids, 40, 1145-1162, (2002) · Zbl 1025.76028 [28] Landau, L.D.; Lifshitz, E.M., Mechanics, (1976), Butterworth-Heinemann London · Zbl 0081.22207 [29] Mellor, G.L.; Blumberg, A.F., Modelling vertical and horizontal diffusivities with the sigma transform system, Appl. Ocean res., 113, 1379-1383, (1985) [30] Miles, J.; Henderson, D., Parametrically forced surface waves, Ann. rev. fluid mech., 22, 143-165, (1990) · Zbl 0723.76004 [31] Ockendon, H.; Ockendon, J.R., Resonant surface waves, J. fluid mech., 59, 397-413, (1973) · Zbl 0273.76006 [32] Perlin, M.; Schultz, W.W., Capillary effects on surface waves, Ann. rev. fluid mech., 32, 241-274, (2000) · Zbl 0988.76013 [33] Phillips, N.A., A coordinate system having some special advantages for numerical forecasting, J. meteorol., 14, 184-185, (1957) [34] Tadjbakhsh, I.; Keller, J.B., Standing surface waves of finite amplitude, J. fluid mech., 8, 442-451, (1960) · Zbl 0173.29402 [35] J.G. Telste, Calculation of fluid motion resulting from large amplitude forced heave motion of a two-dimensiomal cylinder in a free surface, in: Proceedings of the Fourth International Conference on numerical Ship Hydrodynamics, Washington, USA, 1985, pp. 81-93 [36] Tsai, C.-P.; Jeng, D.-S., Numerical Fourier solutions of standing waves in finite water depth, Appl. Ocean res., 16, 185-193, (1994) [37] Turnbull, M.S.; Borthwick, A.G.L.; Eatock Taylor, R., Numerical wave tank based on a σ-transformed finite element inviscid flow solver, Int. J. numer. meth. fluids, 42, 641-663, (2003) · Zbl 1143.76490 [38] Ushijima, S., Three-dimensional arbitrary lagrangian – eulerian numerical prediction method for non-linear free surface oscillation, Int. J. numer. meth. fluids, 26, 605-623, (1998) · Zbl 0927.76075 [39] Vanden-Broeck, J-M.; Schwartz, L.W., Numerical calculation of standing waves in water of arbitrary uniform depth, Phys. fluids, 24, 5, 812-815, (1981) [40] Virnig, J.C.; Berman, A.S.; Sethna, P.R., On three-dimensional nonlinear subharmonic resonance surface waves in a fluid: part ii – experiment, J. appl. mech., 55, 220-224, (1988) [41] Waterhouse, D.D., Resonance sloshing near critical depth, J. fluid mech., 281, 313-318, (1994) · Zbl 0822.76009 [42] Wu, G.X.; Eatock Taylor, R., Finite element analysis of two-dimensional non-linear transient water waves, Appl. Ocean res., 16, 363-372, (1994) [43] Wu, G.X.; Ma, Q.A.; Eatock Taylor, R., Numerical simulation of sloshing waves in a 3D tank based on a finite element method, Appl. Ocean res., 20, 337-355, (1998)
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